<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon">f</font> be   <a href="funct_1.html#NM1">Function</a>;<br/><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>



<b>assume </b><a NAME="E1:63"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><a href="funct_6.html#K3" target="_self">rngs</a> <font color="Maroon">f</font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">g</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E3:63"/><i><font color="Green">E40</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="funct_6.html#K3" target="_self">rngs</a> <font color="Maroon">f</font></span>)</span> &amp; ( for <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="funct_6.html#K3" target="_self">rngs</a> <font color="Maroon">f</font></span>)</span> holds <br/><font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="funct_6.html#K3" target="_self">rngs</a> <font color="Maroon">f</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">y</font> ) )
 <b>by </b><i><a class="ref" href="card_3.html#D5">CARD_3:def 5</a></i>;<br/>
<a NAME="E4:63"/><i><font color="Green">E41</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="funct_6.html#K3" target="_self">rngs</a> <font color="Maroon">f</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="relat_1.html#K10">"</a> <span class="p1">(<span class="default"><a href="funct_6.html#K1" target="_self">SubFuncs</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font></span>)</span></span>)</span> &amp;  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2" target="_self">doms</a> <font color="Maroon">f</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="relat_1.html#K10">"</a> <span class="p1">(<span class="default"><a href="funct_6.html#K1" target="_self">SubFuncs</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font></span>)</span></span>)</span> &amp;  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2" target="_self">doms</a> <font color="Maroon">f</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="funct_6.html#K7" target="_self">Frege</a> <font color="Maroon">f</font></span>)</span> )
 
<b>by </b><i>, , </i>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="hidden.html#M1">set</a> ) <b>-&gt; </b>   <a href="hidden.html#M1">set</a>  = <span class="p1">(<span class="default"><a href="funct_6.html#K2" target="_self">doms</a> <font color="Maroon">f</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b> ex <font color="Olive">f1</font> being  <a href="funct_1.html#NM1">Function</a> st <br/>( <font color="Olive">f1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> &amp; <font color="Olive">f1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> );<br/>
<b>consider </b><font color="Maroon">h</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E6:63"/><i><font color="Green">E42</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">h</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="relat_1.html#K10">"</a> <span class="p1">(<span class="default"><a href="funct_6.html#K1" target="_self">SubFuncs</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font></span>)</span></span>)</span> &amp; ( for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="relat_1.html#K10">"</a> <span class="p1">(<span class="default"><a href="funct_6.html#K1" target="_self">SubFuncs</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font></span>)</span></span>)</span> holds <br/>for <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> iff ( <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">x</font>) &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Olive">y</font>] ) ) ) )
 <b>from </b><i><a class="ref" href="card_3.html#S4">CARD_3:sch 4</a>();<br/></i>
<div><i><font color="Green">E43</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="63_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">X</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:63_1"/>
<font color="Maroon">X</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">h</font>
 ;<br/><b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:63_1"/><i><font color="Green">E52</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">h</font> &amp; <font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E4:63_1"/><i><font color="Green">E53</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="funct_6.html#K1" target="_self">SubFuncs</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font></span>)</span> )
 <b>by </b><i><a class="txt" href="funct_6.html#E6:63"><i><font color="Green">E42</font></i></a>, <a class="ref" href="funct_6.html#T2" target="_self">Th2</a>, <a class="ref" href="funct_1.html#D13">FUNCT_1:def 13</a></i>;<br/><b>then reconsider </b><font color="Maroon">fx</font> = <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> as   <a href="funct_1.html#NM1">Function</a> <b>by </b><i/>;<br/><a NAME="E6:63_1"/>
( <span class="p1">(<span class="default"><a href="funct_6.html#K3" target="_self">rngs</a> <font color="Maroon">f</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">fx</font> &amp; <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="funct_6.html#K3" target="_self">rngs</a> <font color="Maroon">f</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> )
 <b>by </b><i><a class="txt" href="funct_6.html#E3:63"><i><font color="Green">E40</font></i></a>, <a class="txt" href="funct_6.html#E4:63"><i><font color="Green">E41</font></i></a>, <a class="txt" href="funct_6.html#E6:63"><i><font color="Green">E42</font></i></a>, <a class="ref" href="funct_6.html#T2" target="_self">Th2</a>, <a class="ref" href="funct_6.html#T3" target="_self">Th3</a>, </i>;<br/><b>then consider </b><font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E8:63_1"/><i><font color="Green">E54</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">fx</font> &amp; <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">fx</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E9:63_1"/>
<span class="p1">(<span class="default"><a href="funct_6.html#K2" target="_self">doms</a> <font color="Maroon">f</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">fx</font>
 <b>by </b><i><a class="ref" href="funct_6.html#T3" target="_self">Th3</a>, </i>;<br/><b>hence </b><a NAME="E10:63_1"/>
<font color="Maroon">X</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="txt" href="funct_6.html#E6:63"><i><font color="Green">E42</font></i></a>, <a class="ref" href="funct_6.html#T2" target="_self">Th2</a>, <a class="ref" href="funct_6.html#T5" target="_self">Th5</a></i>;<br/></div><b>end;</b></div>
<div><i><font color="Green">E55</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="63_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:63_2"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">f</font> <a href="relat_1.html#K10">"</a> <span class="p1">(<span class="default"><a href="funct_6.html#K1" target="_self">SubFuncs</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><a NAME="E2:63_2"/><b>then </b><i><font color="Green">E57</font></i>: 
( <font color="Maroon">g</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  &amp;  <a href="funct_6.html#K2" target="_self">doms</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 <b>by </b><i><a class="txt" href="funct_6.html#E3:63"><i><font color="Green">E40</font></i></a>, <a class="txt" href="funct_6.html#E4:63"><i><font color="Green">E41</font></i></a>, <a class="ref" href="relat_1.html#T64">RELAT_1:64</a></i>;<br/><a NAME="E3:63_2"/>
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="funct_6.html#K7" target="_self">Frege</a> <font color="Maroon">f</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="txt" href="funct_6.html#E4:63"><i><font color="Green">E41</font></i></a>, <a class="ref" href="funct_6.html#T3" target="_self">Th3</a>, <a class="ref" href="card_3.html#T19">CARD_3:19</a>, <a class="ref" href="relat_1.html#T64">RELAT_1:64</a></i>;<br/><a NAME="E4:63_2"/><b>then </b><i><font color="Green">E58</font></i>: 
 <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="funct_6.html#K7" target="_self">Frege</a> <font color="Maroon">f</font></span>)</span>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>then consider </b><font color="Maroon">h</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E6:63_2"/><i><font color="Green">E59</font></i>: 
( <span class="p1">(<span class="default"><a href="funct_6.html#K7" target="_self">Frege</a> <font color="Maroon">f</font></span>)</span> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">h</font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">h</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="relat_1.html#K10">"</a> <span class="p1">(<span class="default"><a href="funct_6.html#K1" target="_self">SubFuncs</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font></span>)</span></span>)</span> &amp; ( for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">h</font> holds <br/><font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="funct_5.html#K4">uncurry</a> <font color="Maroon">f</font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Olive">x</font>,<span class="p1">(<span class="default"><a href="xboole_0.html#K1">{}</a>  <a href="funct_1.html#K1">.</a> <font color="Olive">x</font></span>)</span> ) )
 <b>by </b><i><a class="txt" href="funct_6.html#E4:63"><i><font color="Green">E41</font></i></a>, </i>;<br/><a NAME="E7:63_2"/>
<font color="Maroon">h</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="funct_6.html#T3" target="_self">Th3</a>, <a class="ref" href="funct_6.html#T10" target="_self">Th10</a>, <a class="ref" href="relat_1.html#T64">RELAT_1:64</a></i>;<br/><b>hence </b><a NAME="E8:63_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="funct_6.html#K7" target="_self">Frege</a> <font color="Maroon">f</font></span>)</span>
 <b>by </b><i><a class="txt" href="funct_6.html#E3:63"><i><font color="Green">E40</font></i></a>, <a class="ref" href="funct_6.html#T5" target="_self">Th5</a>, <a class="ref" href="funct_6.html#T7" target="_self">Th7</a>, <a class="ref" href="funct_6.html#T10" target="_self">Th10</a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="63_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:63_3"/>
<font color="Maroon">f</font> <a href="relat_1.html#K10">"</a> <span class="p1">(<span class="default"><a href="funct_6.html#K1" target="_self">SubFuncs</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><b>then reconsider </b><font color="Maroon">D</font> =  <a href="relat_1.html#K2">rng</a> <font color="Maroon">h</font> as  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>  <b>by </b><i><a class="txt" href="funct_6.html#E6:63"><i><font color="Green">E42</font></i></a>, <a class="ref" href="relat_1.html#T65">RELAT_1:65</a></i>;<br/><b>consider </b><font color="Maroon">Ch</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E4:63_3"/><i><font color="Green">E60</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">Ch</font> <a href="hidden.html#R1">=</a> <font color="Maroon">D</font> &amp; ( for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font> holds <br/><font color="Maroon">Ch</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">x</font> ) )
 <b>by </b><i><a class="txt" href="funct_6.html#E7:63"><i><font color="Green">E43</font></i></a>, <a class="ref" href="wellord2.html#T28">WELLORD2:28</a></i>;<br/><a NAME="E5:63_3"/><i><font color="Green">E61</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">Ch</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">h</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">h</font>
 <b>by </b><i><a class="ref" href="funct_6.html#T3" target="_self">Th3</a>, <a class="ref" href="relat_1.html#T46">RELAT_1:46</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="63_3_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:63_3_1"/><i><font color="Green">E88</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2" target="_self">doms</a> <font color="Maroon">f</font></span>)</span>
 ;<br/><a NAME="E2:63_3_1"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon">Ch</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">h</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Ch</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font></span>)</span> &amp; <font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">h</font> )
 <b>by </b><i><a class="txt" href="funct_6.html#E4:63"><i><font color="Green">E41</font></i></a>, <a class="txt" href="funct_6.html#E6:63"><i><font color="Green">E42</font></i></a>, <a class="ref" href="funct_6.html#T5" target="_self">Th5</a>, <a class="ref" href="funct_1.html#T22">FUNCT_1:22</a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E3:63_3_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">Ch</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">h</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font>
 <b>by </b><i><a class="ref" href="funct_6.html#T3" target="_self">Th3</a></i>;<br/><b>hence </b><a NAME="E4:63_3_1"/>
<span class="p1">(<span class="default"><font color="Maroon">Ch</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">h</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2" target="_self">doms</a> <font color="Maroon">f</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font>
 <b>by </b><i><a class="txt" href="funct_6.html#E4:63"><i><font color="Green">E41</font></i></a>, <a class="txt" href="funct_6.html#E6:63"><i><font color="Green">E42</font></i></a>, <a class="ref" href="funct_6.html#T7" target="_self">Th7</a></i>;<br/></div><b>end;</b></div><a NAME="E7:63_3"/><b>then </b><i><font color="Green">E89</font></i>: 
<font color="Maroon">Ch</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">h</font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2" target="_self">doms</a> <font color="Maroon">f</font></span>)</span>
 <b>by </b><i><a class="txt" href="funct_6.html#E4:63"><i><font color="Green">E41</font></i></a>, <a class="txt" href="funct_6.html#E6:63"><i><font color="Green">E42</font></i></a>, <a class="ref" href="funct_6.html#T5" target="_self">Th5</a>, <a class="ref" href="card_3.html#D5">CARD_3:def 5</a></i>;<br/><b>then consider </b><font color="Maroon">h1</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E9:63_3"/><i><font color="Green">E90</font></i>: 
( <span class="p1">(<span class="default"><a href="funct_6.html#K7" target="_self">Frege</a> <font color="Maroon">f</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">Ch</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">h</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">h1</font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">h1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="relat_1.html#K10">"</a> <span class="p1">(<span class="default"><a href="funct_6.html#K1" target="_self">SubFuncs</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font></span>)</span></span>)</span> &amp; ( for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">h1</font> holds <br/><font color="Maroon">h1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="funct_5.html#K4">uncurry</a> <font color="Maroon">f</font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Olive">x</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">Ch</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">h</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font></span>)</span> ) )
 <b>by </b><i/>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="63_3_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">z</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:63_3_2"/><i><font color="Green">E91</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="relat_1.html#K10">"</a> <span class="p1">(<span class="default"><a href="funct_6.html#K1" target="_self">SubFuncs</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font></span>)</span></span>)</span>
 ;<br/><a NAME="E2:63_3_2"/><b>then </b><i><font color="Green">E92</font></i>: 
( <font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> &amp; <font color="Maroon">h1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="funct_5.html#K4">uncurry</a> <font color="Maroon">f</font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">z</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">Ch</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">h</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">Ch</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">h</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Ch</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font></span>)</span> &amp; <font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">h</font> )
 <b>by </b><i><a class="txt" href="funct_6.html#E6:63"><i><font color="Green">E42</font></i></a>, <a class="ref" href="funct_6.html#T5" target="_self">Th5</a>, <a class="ref" href="funct_6.html#T10" target="_self">Th10</a>, <a class="ref" href="funct_1.html#T22">FUNCT_1:22</a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a>, <a class="ref" href="funct_1.html#D13">FUNCT_1:def 13</a></i>;<br/><a NAME="E3:63_3_2"/><b>then </b><i><font color="Green">E107</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">Ch</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">h</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font>
 <b>by </b><i><a class="ref" href="funct_6.html#T3" target="_self">Th3</a></i>;<br/><b>then consider </b><font color="Maroon">f1</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E5:63_3_2"/><i><font color="Green">E108</font></i>: 
( <font color="Maroon">f1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> &amp; <font color="Maroon">f1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">Ch</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">h</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> )
 <b>by </b><i><a class="txt" href="funct_6.html#E6:63"><i><font color="Green">E42</font></i></a>, <a class="ref" href="funct_6.html#T11" target="_self">Th11</a></i>;<br/><a NAME="E6:63_3_2"/>
( <span class="p1">(<span class="default"><font color="Maroon">Ch</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">h</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2" target="_self">doms</a> <font color="Maroon">f</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> &amp; <span class="p1">(<span class="default"><a href="funct_6.html#K2" target="_self">doms</a> <font color="Maroon">f</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f1</font> )
 <b>by </b><i><a class="txt" href="funct_6.html#E6:63"><i><font color="Green">E42</font></i></a>, <a class="ref" href="funct_6.html#T11" target="_self">Th11</a>, <a class="ref" href="funct_6.html#T12" target="_self">Th12</a>, <a class="ref" href="funct_6.html#T13" target="_self">Th13</a>, , </i>;<br/><b>hence </b><a NAME="E7:63_3_2"/>
<font color="Maroon">h1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font>
 <b>by </b><i><a class="ref" href="funct_6.html#T12" target="_self">Th12</a>, , <a class="ref" href="funct_5.html#T45">FUNCT_5:45</a></i>;<br/></div><b>end;</b></div><a NAME="E11:63_3"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h1</font>
 <b>by </b><i><a class="txt" href="funct_6.html#E3:63"><i><font color="Green">E40</font></i></a>, <a class="txt" href="funct_6.html#E4:63"><i><font color="Green">E41</font></i></a>, <a class="ref" href="funct_6.html#T10" target="_self">Th10</a>, <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/><b>hence </b><a NAME="E12:63_3"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="funct_6.html#K7" target="_self">Frege</a> <font color="Maroon">f</font></span>)</span>
 <b>by </b><i><a class="txt" href="funct_6.html#E4:63"><i><font color="Green">E41</font></i></a>, <a class="ref" href="funct_6.html#T7" target="_self">Th7</a>, <a class="ref" href="funct_6.html#T10" target="_self">Th10</a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E10:63"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="funct_6.html#K7" target="_self">Frege</a> <font color="Maroon">f</font></span>)</span>
 <b>by </b><i><a class="ref" href="funct_6.html#T2" target="_self">Th2</a></i>;<br/>


</div>
